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1. • CommentRowNumber1.
   • CommentAuthorSamuel Adrian Antz
   • CommentTimeJun 28th 2024
   • (edited Jun 28th 2024)
   • PermaLink
   Author: Samuel Adrian Antz
   Format: MarkdownItexCreated basic outline with some important connections. Yang-Mills measure, after all the main concept which makes this special case interesting, and references will be added later. Edit: Crosslinked [[D=2 Yang-Mills theory]] on related pages: [[D=2 QCD]], [[D=4 Yang-Mills theory]], [[D=5 Yang-Mills theory]]. <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

   Created basic outline with some important connections. Yang-Mills measure, after all the main concept which makes this special case interesting, and references will be added later.

   Edit: Crosslinked D=2 Yang-Mills theory on related pages: D=2 QCD, D=4 Yang-Mills theory, D=5 Yang-Mills theory.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2024
   • (edited Jun 28th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexwhere you had a link to *[[quantization]]* I made it point to *[[charge quantization]]* instead where it said "[[Chern-Weil theory]] implies" I expanded to "[[Chern-Weil theory]] implies that the [[first Chern class]] of the gauge bundle is:" the section "Application to the 2-sphere" is lacking, I find, a sentence connecting back to the previous section. To this end, I have given your first display math environment the label "`FirstChernClassByIntegration`", so that you can refer to this by typing "`(eq:FirstChernClassByIntegration)`" apart from this I made some cosmetic edits here and there beware that our Instiki parser demands more whitespace than ordinary LaTeX: it renders "`d_AA`" like "`d_\mathrm{AA}`" --- one needs to type "`d_A A`" to get the intended result (which may be a nuisance, but the latter form is also easier to parse by humans) <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/4">v4</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

   where you had a link to quantization I made it point to charge quantization instead

   where it said “Chern-Weil theory implies” I expanded to “Chern-Weil theory implies that the first Chern class of the gauge bundle is:”

   the section “Application to the 2-sphere” is lacking, I find, a sentence connecting back to the previous section. To this end, I have given your first display math environment the label “FirstChernClassByIntegration”, so that you can refer to this by typing “(eq:FirstChernClassByIntegration)”

   apart from this I made some cosmetic edits here and there

   beware that our Instiki parser demands more whitespace than ordinary LaTeX:

   it renders “d_AA” like “d_\mathrm{AA}” — one needs to type “d_A A” to get the intended result (which may be a nuisance, but the latter form is also easier to parse by humans)

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorSamuel Adrian Antz
   • CommentTimeJun 28th 2024
   • (edited Jun 28th 2024)
   • PermaLink
   Author: Samuel Adrian Antz
   Format: MarkdownItexThanks for the additions! I've also noticed "d_AA", but then you already started editing the page. The section of the sphere will definitely get some expansion and connection (pun not intended). But what was wrong about putting [[D=4 Yang-Mills theory]] and [[D=5 Yang-Mills theory]] into related concepts?

   Thanks for the additions! I’ve also noticed “d_AA”, but then you already started editing the page. The section of the sphere will definitely get some expansion and connection (pun not intended). But what was wrong about putting D=4 Yang-Mills theory and D=5 Yang-Mills theory into related concepts?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2024
   • (edited Jun 28th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI didn't remove anything from "Related concepts". Maybe our edits confliced, and I accidentally overwrote an edit of yours, sorry. Please put it back in, if you can.

   I didn’t remove anything from “Related concepts”. Maybe our edits confliced, and I accidentally overwrote an edit of yours, sorry. Please put it back in, if you can.

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 28th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## References The original reference: * [[A. Migdal]], _Recursion equations in gauge field theories_, Zh. Eksp. Teor. Fiz. 69 (1975), 810–822 (Sov. Phys. JETP 42, 413–418). [PDF](http://jetp.ras.ru/cgi-bin/dn/e_042_03_0413.pdf). Further developments: * Ninoslav E. Bralić, _Exact computation of loop averages in two-dimensional Yang-Mills theory_, Phys. Rev. D 22:12 (1980), 3090–3103. [doi:10.1103/PhysRevD.22.3090](https://doi.org/10.1103/PhysRevD.22.3090). * V. A. Kazakov, I. K. Kostov, _Non-linear strings in two-dimensional U(∞) gauge theory_, Nuclear Physics B 176:1 (1980), 199–215. [doi][1] [1]: https://doi.org/10.1016/0550-3213(80)90072-3 * Leonard Gross, Christopher King, Ambar Sengupta, _Two dimensional Yang-Mills theory via stochastic differential equations_, Annals of Physics 194:1 (1989), 65–112. [doi][1] [1]: https://doi.org/10.1016/0003-4916(89)90032-8 * B. Ye. Rusakov, _LOOP AVERAGES AND PARTITION FUNCTIONS IN U(N) GAUGE THEORY ON TWO-DIMENSIONAL MANIFOLDS_, Modern Physics Letters A 5:9 (1990), 693–703. [doi](https://doi.org/10.1142/S0217732390000780). * Dana S. Fine, _Quantum Yang-Mills on the two-sphere_, Communications in Mathematical Physics 134 (1990), 273–292. [doi](https://doi.org/10.1007/BF02097703). * Dana S. Fine, _Quantum Yang-Mills on a Riemann surface_, Communications in Mathematical Physics 140 (1991), 321–338. [doi](https://doi.org/10.1007/BF02099502). * Ambar Sengupta, _The Yang-Mills measure for $S^2$_, Journal of Functional Analysis 108:2 (1992), 231–273. [doi][1] [1]: https://doi.org/10.1016/0022-1236(92)90025-E * Ambar Sengupta, _Quantum Gauge Theory on Compact Surfaces_, Annals of Physics 221:1 (1993), 17–52. [doi](https://doi.org/10.1006/aphy.1993.1002) * [[Matthias Blau]], George Thompson, _QUANTUM YANG-MILLS THEORY ON ARBITRARY SURFACES_, International Journal of Modern Physics A 7:16 (1992), 3781–3806. [doi](https://doi.org/10.1142/S0217751X9200168X). <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

   Added:

   References

   The original reference:

   • A. Migdal, Recursion equations in gauge field theories, Zh. Eksp. Teor. Fiz. 69 (1975), 810–822 (Sov. Phys. JETP 42, 413–418). PDF.

   Further developments:

   • Ninoslav E. Bralić, Exact computation of loop averages in two-dimensional Yang-Mills theory, Phys. Rev. D 22:12 (1980), 3090–3103. doi:10.1103/PhysRevD.22.3090.

   • V. A. Kazakov, I. K. Kostov, Non-linear strings in two-dimensional U(∞) gauge theory, Nuclear Physics B 176:1 (1980), 199–215. doi

   • Leonard Gross, Christopher King, Ambar Sengupta, Two dimensional Yang-Mills theory via stochastic differential equations, Annals of Physics 194:1 (1989), 65–112. doi

   • B. Ye. Rusakov, LOOP AVERAGES AND PARTITION FUNCTIONS IN U(N) GAUGE THEORY ON TWO-DIMENSIONAL MANIFOLDS, Modern Physics Letters A 5:9 (1990), 693–703. doi.

   • Dana S. Fine, Quantum Yang-Mills on the two-sphere, Communications in Mathematical Physics 134 (1990), 273–292. doi.

   • Dana S. Fine, Quantum Yang-Mills on a Riemann surface, Communications in Mathematical Physics 140 (1991), 321–338. doi.

   • Ambar Sengupta, The Yang-Mills measure for $S^2$, Journal of Functional Analysis 108:2 (1992), 231–273. doi

   • Ambar Sengupta, Quantum Gauge Theory on Compact Surfaces, Annals of Physics 221:1 (1993), 17–52. doi

   • Matthias Blau, George Thompson, QUANTUM YANG-MILLS THEORY ON ARBITRARY SURFACES, International Journal of Modern Physics A 7:16 (1992), 3781–3806. doi.

   diff, v6, current

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 28th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexOne more: * [[Edward Witten]], _Two-dimensional gauge theories revisited_, Journal of Geometry and Physics 9:4 (1992), 303–368. [arXiv:hep-th/9204083](https://arxiv.org/abs/hep-th/9204083). [doi:10.1016/0393-0440(92)90034-X](https://doi.org/10.1016/0393-0440(92)90034-X). <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

   One more:

   • Edward Witten, Two-dimensional gauge theories revisited, Journal of Geometry and Physics 9:4 (1992), 303–368. arXiv:hep-th/9204083. doi:10.1016/0393-0440(92)90034-X.

   diff, v6, current

7. • CommentRowNumber7.
   • CommentAuthorperezl.alonso
   • CommentTimeJun 30th 2024
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexmore pointers * [[Edward Witten]]. On quantum gauge theories in two dimensions. *Commun.Math. Phys*. 141, 153–209 (1991). ([doi](https://doi.org/10.1007/BF02100009)). * Stefan Cordes, [[Gregory Moore]], [[Sanjaye Ramgoolam]]. *Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories* (1994). ([hep-th/9411210](https://arxiv.org/abs/hep-th/9411210)). * [[David Gross]]. *Two Dimensional QCD as a String Theory* (1992). ([hep-th/9212149](https://arxiv.org/abs/hep-th/9212149)) * [[David Gross]], [[Washington Taylor]], “Two-dimensional QCD is a string theory,” Nucl. Phys. B400 (1993). ([hep-th/9301068](https://arxiv.org/abs/hep-th/9301068)). * [[David Gross]], [[Washington Taylor]], “Twists and Wilson loops in the string theory of two-dimensional QCD,” Nucl. Phys. B 403 (1993) 395-452. ([hep-th/9303046](https://arxiv.org/abs/hep-th/9303046)). * [[Petr Hořava]]. *Topological Strings and QCD in Two Dimensions* (1993). ([hep-th/9311156](https://arxiv.org/abs/hep-th/9311156)). * [[Petr Hořava]]. *Topological Rigid String Theory and Two Dimensional QCD* (1995). ([hep-th/9507060](https://arxiv.org/abs/hep-th/9507060)). * Mendel Nguyen, Yuya Tanizaki, Mithat Ünsal. *Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory* (2021). ([arXiv:2104.01824](https://arxiv.org/abs/2104.01824)). * [[Tony Pantev]], [[Eric Sharpe]]. *Decomposition and the Gross-Taylor string theory* (2023). ([arXiv:2307.08729](https://arxiv.org/abs/2307.08729)). * Richard Wedeen. *Volume-Dependent Field Theories* (2024). ([arXiv:2402.06691](https://arxiv.org/abs/2402.06691)). <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/8">v8</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

   more pointers

   • Edward Witten. On quantum gauge theories in two dimensions. Commun.Math. Phys. 141, 153–209 (1991). (doi).

   • Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam. Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories (1994). (hep-th/9411210).

   • David Gross. Two Dimensional QCD as a String Theory (1992). (hep-th/9212149)

   • David Gross, Washington Taylor, “Two-dimensional QCD is a string theory,” Nucl. Phys. B400 (1993). (hep-th/9301068).

   • David Gross, Washington Taylor, “Twists and Wilson loops in the string theory of two-dimensional QCD,” Nucl. Phys. B 403 (1993) 395-452. (hep-th/9303046).

   • Petr Hořava. Topological Strings and QCD in Two Dimensions (1993). (hep-th/9311156).

   • Petr Hořava. Topological Rigid String Theory and Two Dimensional QCD (1995). (hep-th/9507060).

   • Mendel Nguyen, Yuya Tanizaki, Mithat Ünsal. Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory (2021). (arXiv:2104.01824).

   • Tony Pantev, Eric Sharpe. Decomposition and the Gross-Taylor string theory (2023). (arXiv:2307.08729).

   • Richard Wedeen. Volume-Dependent Field Theories (2024). (arXiv:2402.06691).

   diff, v8, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexNotice the separate entry *[[D=2 QCD]]*.

   Notice the separate entry D=2 QCD.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2024
   • (edited Jun 30th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFixed the DOI-links introduced in [#5](#Comment_117768): When you use the syntax [doi][1] [1]: http:... you need of course to replace "1" by a distinct number every time the command appears, or else they will all produce the same hyperlink (see the [[Sandbox]]) <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/9">v9</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

   Fixed the DOI-links introduced in #5: When you use the syntax

     [doi][1] [1]: http:...
   

   you need of course to replace “1” by a distinct number every time the command appears, or else they will all produce the same hyperlink (see the Sandbox)

   diff, v9, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2024
    • (edited Jun 30th 2024)
    • PermaLink
    Author: Urs
    Format: MarkdownItexHave tried to brush-up the list of references (please don't use all capitals) and to harmonize it with those at *[[D=2 QCD]]* <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/9">v9</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

    Have tried to brush-up the list of references (please don’t use all capitals) and to harmonize it with those at D=2 QCD

    diff, v9, current

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's * [[Marco Benini]], Tomás Fernández, [[Alexander Schenkel]]: *Derived algebraic geometry of 2d lattice Yang-Mills theory* &lbrack;[arXiv:2409.06873](https://arxiv.org/abs/2409.06873)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/D%3D2+Yang-Mills+theory/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D2+Yang-Mills+theory/10">v10</a>, <a href="https://ncatlab.org/nlab/show/D%3D2+Yang-Mills+theory">current</a>

    added pointer to today’s

    • Marco Benini, Tomás Fernández, Alexander Schenkel: Derived algebraic geometry of 2d lattice Yang-Mills theory [arXiv:2409.06873]

    diff, v10, current